Exam 16: Analysis of Variance

Provide an example for a randomised block design with three treatments (k = 3) and four blocks (b = 4), in which SST is equal to zero and SSB and SSE are not equal to zero.

Conceptually and mathematically, the F-test of the independent-samples single-factor ANOVA is an extension of the t-test of $\mu _ { 1 } - \mu _ { 2 }$ .

A pharmaceutical manufacturer has been researching new formulas to provide quicker relief of minor pains. His laboratories have produced three different formulas, which he wanted to test. Fifteen people who complained of minor pains were recruited for an experiment. Five were given formula 1, five were given formula 2, and the last five were given formula 3. Each was asked to take the medicine and report the length of time until some relief was felt. The results are shown below. Do these data provide sufficient evidence to indicate that differences in the time of relief exist among the three formulas? Use $\alpha$ = 0.05. Time in minutes until reliefis felt Formula 1 Formula 2 Formula 3 4 2 6 8 5 7 6 3 7 9 7 8 8 1 6

Consider the following ANOVA table: Source of Variation SS df MS F Treatments 4 2 2.0 0.80 Error 30 12 2.5 Total 34 14 The number of treatments is: A. 13. B. 5. C. 3. D. 33.

In a completely randomised design, 12 experimental units were assigned to the first treatment, 15 units to the second treatment, and 18 units to the third treatment. A partial ANOVA table is shown below: Source of Variation SS df MS F Treatments * * * 9 Error * * 35 Total * * Test at the 5% significance level to determine if differences exist among the three treatment means.

Which of the following is not true of Tukey's multiple comparison method? A. It is based on the studentised range statistic q to obtain the critical value needed to construct individual confidence intervals. B. It requires that all sample sizes are equal, or at least similar. C. It can be employed instead of the analysis of variance. D. All of these choices are correct.

The following table shows the average weekly losses of worker hours due to accidents in 2009 at five randomly selected manufacturing firms in New South Wales and at five randomly selected manufacturing firms in Victoria. NSW Victoria 45 57 73 83 46 34 124 26 33 17 Assume that the weekly losses of worker hours are normally distributed. Perform an equal-variances t-test at the 5% significance level to determine whether the population means differ.

A financial analyst studied the percentage rates of return of three different types of mutual funds. Random samples of percentage rates of return for five periods were taken from each fund. The results appear in the table below. Mutual Funds Percentage Rates Fund A Fund B Fund C 12 4 9 15 8 3 13 6 5 14 5 7 17 4 4 Test at the 5% significance level to determine whether the mean percentage rates for the three funds differ, assuming that the distribution is normal.

The F-test of the randomised block design of the analysis of variance has the same requirements as the independent-samples design; that is, the random variable must be normally distributed and the population variances must be equal.

A recent college graduate is in the process of deciding which one of three US graduate schools he should apply to. He decides to judge the quality of the schools on the basis of the Graduate Management Admission Test (GMAT) scores of those who are accepted into the school. A random sample of six students in each school produced the following GMAT scores. GMAT Scores School 1 School 2 School 3 650 510 590 620 550 510 630 700 520 580 630 500 710 600 490 690 650 530 Assuming that the data are normally distributed, can he infer at the 10% significance level that the GMAT scores differ among the three schools?

A statistician employed by a television rating service wanted to determine whether there were differences in television viewing habits among three residential areas in South Australia. He took a random sample of five adults in each of the areas and asked each to report the number of hours spent watching television in the previous week. From the data shown below, can he infer at the 5% level of significance that differences in hours of television watching exist among the three residential areas? Henley Beach West Lakes Shore Glenelg 25 30 23 25 33 18 18 40 21 15 29 20 27 36 10

In one-way ANOVA, suppose that there are four treatments with n₁ = 7, n₂ = 6, n₃ = 5, and n₄ = 7. Then the rejection region for this test at the 1% level of significance is: A. F> B. F> C. F> D. F>

Which of the following is compared in ANOVA ? A. Two or more population standard deviations. B. Two or more population means. C. Two or more population variances. D. All of these choices are correct.

Two samples of 10 each have been taken from the male and female workers of a large company. The data involve the wage rate of each worker. To test whether there is any difference in the average wage rate between male and female workers, a pooled-variances t-test will be considered. Another test option to consider is ANOVA. The most likely ANOVA to fit this test situation is one way ANOVA.

The calculated value of F in a one-way analysis is 7.88. The numbers of degrees of freedom for numerator and denominator are 3 and 9, respectively. The most accurate statement to be made about the p-value is that p-value < 0.01.

One-way ANOVA is applied to independent samples taken from three normally distributed populations with equal variances. The following summary statistics are calculated: $n _ { 1 } =$ 10, $\bar { x } _ { 1 } =$ 40, $s _ { 1 } =$ 5. $n _ { 2 } =$ 10, $\bar { x } _ { 2 } =$ 48, $s _ { 2 } =$ 6. $n _ { 3 } =$ 10, $\bar { x } _ { 3 } =$ 50, $s _ { 3 } =$ 4. The between-treatments variation equals: A. 460. B. 688. C. 560. D. 183.

The number of degrees of freedom for the denominator of a one-way ANOVA test for 5 population means with 15 observations sampled from each population is: A. 59. B. 55. C. 56. D. 10.

A statistics course coordinator is trying to improve the teaching quality of her course. Last semester, her class list was randomly divided into three separate groups, A, B and C. Group A are offered two, one hour lectures per week and one, one-hour tutorial per week, where the maximum number in a tutorial class is 10, and their continuous assessment is individual based. Group B are offered two one hour lectures per week and one two hour tutorial per week where the average tutorial class size is 42 and the continuous assessment is team based, where students are randomly allocated to a team of 7 students each in their tutorial class. Group C have one four hour lecture including combined tutorial per week and their continuous assessment is individual based, where this class size is 50 students. A random sample of students from each of group A, B and C was selected with their final marks given in the following table: Group A Group B Group C 75 50 50 68 63 65 99 45 81 72 55 58 52 75 75 a. Is there significant evidence at the 10% level of significance for the course coordinator to infer that there exists a difference in the population mean final mark of the three different teaching methods? b. What assumption about the distribution of final grades was needed in order to carry out the F-test of the analysis of variance?

In ANOVA, error variability is computed as the sum of the squared errors, SSE, for all values of the response variable. This variability is the: A. total variation. B. within-group variation. C. between-groups variation. D. None of these choices are correct.

If we examine two or more independent samples to determine if their population means could be equal, we are performing one-way analysis of variance (ANOVA).